arXiv subjects
Publications and source records attributed to Nikolay A. Ivanov.
We introduce and study quantum phase operators associated with the Quantum Harmonic Oscillator (QHO). We show that these operators are trace-class perturbations of the Susskind-Glogower operators and examine their mathematical and physical properties. The construction is motivated by the physically relevant two-phase case.
We introduce two families of examples of groups acting on trees, one consisting of group amalgamations and the other consisting of HNN-extensions, motivated by the problems of $C^*$-simplicity and unique trace property. Moreover, we prove that our examples are not inner amenable and identify a relatively large, simple, normal subgroup in each one.
We study non-ascending HNN extensions acting on their Bass-Serre trees, and characterize C*-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not C*-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning C*-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.
We give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group C*-algebra, and provide a concrete example of an amalgam with trivial kernel, such that its reduced group C*-algebra has a unique tracial state, but is not simple. Moreover, we show that there is a radical class of groups for which the reduced group C*-algebra of any group is simple precisely when the group has a trivial radical corresponding to this class.
We make some comments on noncommutative $U(N)$-instantons on $\mathbb{R}^4_θ$. We elaborate on the equations for the ASD-connection for free modules. Further we make some remarks on the computation of the topological index of ADHM instantons.
To a graph $Γ$ one can associate a C^*-algebra $C^*(Γ)$ generated by isometries. Such $C^*$-algebras were studied recently by Crisp and Laca. They are a special case of the Toeplitz C^*-algebras $\mathcal{T}(G, P)$ associated to quasi-latice ordered groups (G, P) introduced by Nica. Crisp and Laca proved that the so called "boundary quotients" $C^*_q(Γ)$ of $C^*(Γ)$ are simple and purely infinite. For a certain class of finite graphs $Γ$ we show that $C^*_q(Γ)$ can be represented as a full corner of a crossed product of an appropriate C^*-subalgebra of $C^*_q(Γ)$ built by using $C^*(Γ')$, where $Γ'$ is a subgraph of $Γ$ with one less vertex, by the group $\mathbb{Z}$. Using induction on the number of the vertices of $Γ$ we show that $C^*_q(Γ)$ are nuclear and belong to the small bootstrap class. This also enables us to use the Pimsner-Voiculescu exact sequence to find their K-theory. Finally we use the Kirchberg-Phillips classification theorem to show that those C^*-algebras are isomorphic to tensor products of $\mathcal{O}_n$ for $1 \leq n \leq \infty$.
We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K_0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the later result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag-Solitar groups BS(m,n).